What should I be integrating? And what will it tell me? Assuming the distributions are over [0,1], are you saying something like P(w) = \int_0^1 P(w<Rcosθ|cosθ) d(cosθ)?
In general, how does one separate the product of two probability distributions with one of them known? Basically, I have the distribution of rcosθ, I know that P(cosθ) = 2/(πsinθ), and I want to find P(r). Wolfram Alpha makes me think that a delta function is involved based on what they say...